Double Set:

A double point of a mapping of a surface into space or a curve
into the plane is a point that is the image of two distinct points of
the surface.

For example, if a circle is mapped into the plane as a figure-8, then
the crossing is a double point. If a surface is mapped into space so
that there is self-intersection, the points of self-intersection are
formed by double-points.

Generically, the double-points of a surface in space form curves. For
an immersion, they will form
closed
curves. These curves are called the double curves or the
double locus of the mapping, and the double set is the
set of points of the surface that map to the double locus.